APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS IN NONLINEAR SECOND ORDER EQUATIONS - ADAPA DURGA MADHURI, P.V. NARESH SAGAR, PINISETTI GANGARAJU (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2021)
In the present paper there is a detailed study for the abstract second order (in time) semi linear differential equation on Cauchy problem in which A and B are (generally unbounded) operators which are linear in a Banach space. This sort of problems rise up frequently with in the study of PDE (partial differential equations). In regular we manage a nonlinear perturbation (likely related to spatial derivatives) through the linear terms, which incorporate higher order spatial derivatives. But opposite to the same old approach of lowering the hassle to a 1st order device in some “energy” norm space, we make the use of the factoring approach. The approach lets in the equation to be written as integral equation which consists a double integral related to the non linearity, reflecting the truth that the equation is 2nd order. There are numerous blessings to this technique with a purpose to be illustrated completely with in the examples. At first, we display that the equation is domestically posed well if the nonlinearity is going to satisfy the local Lipschitz condition (on soaking up linear terms into f, higher order derivatives of the operators A and B want be retained.) For partial differential equations, this offers a great rule of thumb for figuring out if a positive hassle is locally properly posed. Energy strategies may be used to expose worldwide existence. Secondly, this technique applies to problems of each hyperbolic and parabolic.